Conjecture REF  allows us to study the hardness of linear system solving and related linear algebraic problems in the style of fine-grained complexity [1]}. The conjecture implies \(\Omega (n^{\omega })\) -hardness of finding exact solutions to linear systems (Lemma REF ). One could hope to get faster algorithms though when allowed to find an approximate solution to the linear system. Specifically, we consider the following notion of approximation:
